Archve o SID Journal o Industral Engneerng 6(00) -6 Absorbng Markov Chan Models to Determne Optmum Process Target evels n Producton Systems wth Rework and Scrappng Mohammad Saber Fallah Nezhad a, Seyed Tagh Akhavan Nak b,* a Department o Industral Engneerng, Yazd nversty, Yazd, Iran b Department o Industral Engneerng, Shar nversty o Technology, Tehran, Iran Receved May., 00; Revsed 5 Jun., 00; Accepted 3 July., 00 Abstract In ths paper, absorbng Markov chan models are developed to determne the optmum process mean levels or both a sngle-stage and a seral two-stage producton system n whch tems are nspected or conormty wth ther speccaton lmts. When the value o the qualty characterstc o an tem alls below a lower lmt, the tem s scrapped. I t alls above an upper lmt, the tem s reworked. Otherwse, the tem passes the nspecton. Ths low o materal through the producton system can be modeled n an absorbng Markov chan characterzng the uncertanty due to scrappng and reworkng. Numercal examples are provded to demonstrate the applcaton o the proposed model. Keywords: Qualty Inspecton; Rework; Markovan Model; Expected Prot.. Introducton The determnaton o optmum process mean s one o the most mportant decson-makng problems encountered n ndustral applcatons. Consder a certan producton process, where an tem s reworked the value o ts qualty characterstc alls above an upper speccaton lmt and t s scrapped when t alls below a lower speccaton lmt. A dmenson n the surace nshng processes o metals s an example o ths scenaro. In ths stuaton, on the one hand the process mean s set too low, then the proporton o nonconormng tems becomes hgh and the decson maker experences hgh rejecton costs assocated wth nonconormng tems. On the other hand, the process mean s set too hgh, then the proporton o reworkng tems becomes hgh, resultng n a hgher reworkng cost. Ths justes the determnaton o the optmum process mean []. Al-Sultan and Pulak [] proposed a model consderng a producton system wth two stages n seres to nd the optmum mean values wth a lower speccaton lmt and applcaton o a 00% nspecton polcy. Ferrell and Chhoker [4] proposed a method to determne the optmal acceptance samplng plans economcally. Ther Approach s based on the Taguch loss uncton to quanty devatons between a qualty characterstc and ts target * Correspondng author emal: Nak@Shar.edu level. Bowlng et al [] employed a Markovan model n order to maxmze the total prot assocated wth a multstage seral producton system. They tred to determne optmal process target levels, n whch lower and upper speccaton lmts are gven at each stage. In addton, they assumed each qualty characterstc s governed by a normal dstrbuton and screenng (00%) nspecton s perormed. Further, Plla and Chandrasekharan [3] modeled the low o materal through the producton system as an absorbng Markov chan consderng scrappng and reworkng. Ther model promses better system desgn n materal requrement plannng, capacty requrement plannng, and nventory control. In the present research, smlar to Bowlng et al[], the low o a dscrete producton process s modeled nto an absorbng Markov chan. In other words, n ths process, not all tems reach the nal stage due to scrappng and reworkng. Hence, the absorbng Markov chan stochastc process model wll be adopted. The data requred or such a model are: () the probablty whch an tem goes rom one stage o producton to the next, and () the probablty o reworkng and scrappng tems at varous stages. At every stage o producton, the tem s nspected; t does not conorm to ts speccatons, t s ether scrapped or reworked. The reworked tem wll be nspected agan. However, there are two man derences between the current work and the one n Bowlng et al []. The rst www.sid.r
Archve o SID Mohammad Saber Fallah Nezhad et al. / Absorbng Markov Chan Models to Determne Optmum Process relates to the assumpton that n the present work the nonconormng tems are repared n a separate staton. In other words, whle n Bowlng et al [] the nonconormng tems are repared n the man staton, n the current research they are repared n a separate repar staton. The second derence s the ncorrectness o the objectve uncton o the two-stage process n Bowlng et al. (004), where they multpled the sellng prce o an tem (SP ) by ( 4)( 4). Knowng that n the ( ) coecent shows derved Markovan model the 4 the percentage o conormng tems, t s not necessary to multply ( 4) SP by ( 4) agan. In ths research, we revse the objectve uncton o Bowlng et al [], tryng to oer a better soluton. The rest o the paper s organzed as ollows. We rst present the requred notatons n secton. The model development comes next n secton 3. The numercal demonstraton on the applcaton o the proposed methodology s gven n secton 4. Fnally, we conclude the paper n secton 5.. Notatons The requred notatons are: : The upper speccaton lmt n the th stage o the producton,, : The lower speccaton lmt n the th stage o the producton,, p j : The probablty o gong rom state to state j n a sngle step j : The long run probablty o gong rom a nonabsorbng state () to another absorbng state (j) E ( PR ): The expected prot per tem E ( RV ): The expected revenue per tem E ( PC ): The expected processng cost per tem E ( SC ): The expected scrappng cost per tem E ( RC ): The expected reworkng cost per tem SP : The sellng prce o an tem PC : The processng cost o the stage SC : The scrappng cost o the RC : The reworkng cost o the th th th stage stage P : The transton probablty matrx Q : The transton probablty matrx o gong rom a non-absorbng state to another non-absorbng state R : A matrx contanng all probabltes o gong rom a non-absorbng state to another absorbng state (.e., accepted or rejected tem) I : The dentty matrx O : A matrx wth zero elements M : The undamental matrx F : The absorpton probablty matrx 3. Model Development Consder a seral producton system n whch tems are 00% nspected n all stages. The tem s then reworked, accepted or scrapped. As raw materals come nto the producton system and nally go out o t, a state n the Markovan model represents derent condtons o the raw materals,.e., reworkng, scrappng, and acceptng. In other words, an tem can be n one o ts three states modeled by a dscrete random varable X. As tme () t goes on, the random varable X generates a random processx t : t 0. Ths stochastc process wth dscrete state space and dscrete values o the parameter t becomes a dscrete tme (rst order) Markov chan when transton rom one state to the next depends only on the current state. Among the states, some are transent and the others absorbng. A Markov chan wth one or more absorbng states s known as absorbng Markov chan. When an tem s n an absorbng state, t never leaves the state (Plla and Chandrasekharan [3]). The expected prot per tem n the system under consderaton can be expressed as ollows: E PR E RV E PC () E SC E RC Then, n what ollows a sngle-stage producton system s rst modeled. The two-stage modelng comes next. 3.. The sngle-stage system Consder a sngle-stage producton system wth the ollowng states: State : An tem s beng processed by the producton system State : An tem s beng reworked State 3: An tem s accepted to be nshed work State 4: An tem s scrapped. Then, the sngle-step transton probablty matrx can be expressed as: 3 4 0 p p p 0 0 p p 0 0 0 0 0 0 3 4 P 3 4 () 3 4 where p s the probablty o reworkng an tem, p3 s the probablty o acceptng an tem, and p 4 s the www.sid.r
Archve o SID Journal o Industral Engneerng 6(00) -6 probablty o scrappng an tem. Assumng that the qualty characterstc o an tem ollows a normal dstrbuton wth mean and standard devaton, these probabltes can be expressed as (Bowlng et al. 004): x p e dx (3) x 3 p e dx (4) x 4 p e dx (5) Moreover, p 3 and p 4 denote the probabltes o acceptng and scrappng a reworked tem, respectvely and the hstorcal data rom the producton system can be used to determne the value o p 3 and p 4. Note that n order to analyze the transton probablty matrx P n an absorbng Markov chan, we rearranged t n the ollowng orm: I O P= (6) R Q By determnng the undamental matrx M as: - p p M= I-Q 0 0 (7) the absorpton probablty matrx F can then be obtaned as ollows (Bowlng et al. 004): p p3 p4 F=M R 0 p3 p 4 p3 p p3 p4 p p4 p3 p 4 3 4 p3 p 4 where 3 and 4 are the probabltes o acceptng and scrappng an tem. Now, the expected prot per tem o equaton () wll be: 3 E PR SP PC SC p RC 4 (8) (9) Substtutng or 3 we have: = SP SC p p p E PR SP p3 p p3 PC SC p p p RC p 3 3 3 3 (0) RCp PC SC p, p, and p the expected Further, substtutng or 3 4 prot can be wrtten n terms o the cumulatve normal dstrbuton as ollows: RC PC SC EPR SP SC p3 The terms and () are unctons o the decson varable and we desre to determne the optmal value o the process mean so that the objectve uncton n () s maxmzed. Ths can be obtaned usng an ordnary numercal search algorthm, n whch the ntervals are rst parttoned to some sub-ntervals or each o whch the objectve uncton value s determned. Then, the maxmum o these values s the near-optmal soluton. 3.. The Two-stage system Consder a two-stage seral producton system wth the ollowng states, State : An tem s beng processed n the rst stage o the producton process State : An tem s beng reworked n the rst stage o the producton process State 3: An tem s beng processed n the second stage o the producton process State 4: An tem s beng reworked n the second stage o the producton process State 5: An tem s accepted to be nshed work State 6: An tem s scrapped Then, assumng the qualty characterstc o an tem n the second stage ollows a normal dstrbuton wth mean and standard devaton, the sngle-step transton probablty matrx can be expressed as ollows: 3 4 5 6 0 p p 0 0 p 0 0 p 0 0 p 30 0 0 p p p 3 6 3 6 34 35 36 4 0 0 0 0 p45 p46 P () where, 50 0 0 0 0 60 0 0 0 0 3 www.sid.r
Archve o SID Mohammad Saber Fallah Nezhad et al. / Absorbng Markov Chan Models to Determne Optmum Process x (3) p e dx x p e dx (4) 3 x p e dx (5) 6 x (6) p e dx 34 x p e dx 35 (7) x p e dx 36 (8) Moreover, p 3 and p 45 denote the probabltes o acceptng a reworked tem n stage one and two, respectvely. Once agan, the hstorcal data avalable n the producton system can be used to determne the values o p 3 and p 45. The terms,,, and are unctons o the decson varables and that are the process means n stages and, respectvely. Rearrangng the P matrx and applyng the method used or the sngle-stage system results n the ollowng undamental and absorpton matrces: p p3 0 0 p3 0 - M= I-Q 0 0 p34 0 0 0 p p p3 p3 p34 p p3 p3 0 p3 p3p34 0 0 p34 0 0 0 F=M R p p p3 p3 p34 p p3 p3 0 p6 0 3 3 34 0 p p p p 6 0 0 p 34 p35 p 36 0 0 0 p45 p46 p p p p p p p p p p p p p p p p p p p p p 35 3 3 34 45 3 3 6 6 36 3 3 34 46 3 3 p3p35 p3p34 p45 p6 p3p36 p3p34p46 p35 p34p45 p36 p34p46 p45 p46 where 5 and 6 are the probabltes o acceptng and scrappng an tem, and 36 s the probablty o scrappng an tem n stage. p p 5 6 5 6 35 36 45 46 (9) The expected prot s obtaned by determnng the terms n equaton () as ollows. 6 E( RV ) SP (0) 4 www.sid.r
Archve o SID Journal o Industral Engneerng 6(00) -6 E( PC) PC Pr tem s processed n stage PC PC pp 3 p3pc E( SC) Pr tem s scrapped n stage SC Pr tem s scrapped n stage SC SCp6 pp6 SCpp3 p3 36 () () () E( RC) Pr tem s reworked n stage RC Pr tem s reworked n stage RC (3) p RC p p p p RC 34 3 3 Thereore, the expected prot per tem or a two-stage seral producton system s obtaned as E PR 6 SP PC p p3 p3 PC p6 p p6 SC ( 4) p p3 p3 36SC RC p p p p p RC 34 3 3 As stated n the ntroducton, to determne the expected prot o a two-stage seral producton system, Bowlng et al. (004) multpled the sellng prce per tem (SP ) by the long-term probablty o acceptng tems n stage multpled by the probablty o acceptng tems n stage. Ths s not correct because the long-term probablty o acceptng tems n stage denotes the overall proporton o the tems that have been accepted n all stages o the process. As a result, t s not requred to multply t by the probablty o acceptng tems n stage. 4. Numercal Examples In ths secton, we provde two numercal examples to llustrate the applcatons o the proposed model n both sngle-stage and two-stage processes. 4.. A numercal example or a sngle-stage system (4) determned by equaton (3), (4), and (5) usng a numercal search method. Then, the expected prot per tem s maxmzed at 9.9 wth a value o E( PR) $73.. The uncton E ( PR ) dened n equaton () s plotted or derent values o the decson varable. Fgure () shows the expected prot as a concave uncton o the process mean. Expected Prot 80 70 60 50 40 30 0 0 0 8 9 0 Process mean Fg.. The expected prot per tem versus the process mean 4.. A numercal example or a two-stage system Consder a two-stage producton system and the ollowng parameters: SP $0, PC $35, PC $30, RC $30, RC $5, SC $5, SC $,.0,.0, p3 0.95, p45 0.95,, 3,, and 7 8. The other probablty terms are rst determned usng equaton (3) and through a numercal search method. Then, the expected prot s maxmzed at 9.8 and 5 wth an expected prot per tem o E( PR) $9.06. The uncton E ( PR ) that s dened n equaton (5) s plotted or derent decson varables and n Fgure (). Once agan, Fgure () shows that the expected prot s a concave uncton o the process mean. 0 Consder a sngle-stage producton system wth the ollowng characterstcs: SP $0, PC $40, RC $35, SC $5,, p3 0.95, 8, and. Note that p 45 does not exst n a sngle stage model and the other probablty terms are 5 Expected Prot -30-60 -90 7.5 6.0 8 4.5 Process mean 0 3.0 Process mean www.sid.r
Mohammad Saber Fallah Nezhad et al. / Absorbng Markov Chan Models to Determne Optmum Process Archve o SID Fg.. The expected prot per tem versus the process mean 5. Concluson In ths paper, absorbng Markov chan models were developed to determne the optmal process means that maxmze the expected prot per tem o both sngle-stage and two-stage producton systems n whch the tems are %00 nspected to be classed as acceptng, scrappng, and reworkng ones. Two numercal examples were provded to llustrate the applcatons o the proposed models. Fnally, the relatonshps between the process means and the expected prot per tem were gven. 6. Reerences [] Bowlng, S.R., Khasawneh, M.T., Kaewkuekool, S., Cho, B.R. A Markovan approach to determnng optmum process target levels or a mult-stage seral producton system. European Journal o Operatonal Research 59: 636 650, 004. [] Al-Sultan, K.S., Pulak, M.F.S. Optmum target values or two machnes n seres wth 00% nspecton. European Journal o Operatonal Research 0: 8 89, 000. [3] Plla, V.M., Chandrasekharan, M.P. An absorbng Markov chan model or producton systems wth rework and scrappng. Computers & Industral Engneerng 55: 695 706, 008. [4] Ferrell, W.G. Jr., Chhoker, A. Desgn o economcally optmal acceptance samplng plans wth nspecton error. Computers & Operatons Research 9: 83-300, 00. 6 www.sid.r